Problem: Simplify; express your answer in exponential form. Assume $p\neq 0, t\neq 0$. $\dfrac{{(p^{5}t^{2})^{2}}}{{(p^{-1}t^{-3})^{2}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(p^{5}t^{2})^{2} = (p^{5})^{2}(t^{2})^{2}}$ On the left, we have ${p^{5}}$ to the exponent ${2}$ . Now ${5 \times 2 = 10}$ , so ${(p^{5})^{2} = p^{10}}$ Apply the ideas above to simplify the equation. $\dfrac{{(p^{5}t^{2})^{2}}}{{(p^{-1}t^{-3})^{2}}} = \dfrac{{p^{10}t^{4}}}{{p^{-2}t^{-6}}}$ Break up the equation by variable and simplify. $\dfrac{{p^{10}t^{4}}}{{p^{-2}t^{-6}}} = \dfrac{{p^{10}}}{{p^{-2}}} \cdot \dfrac{{t^{4}}}{{t^{-6}}} = p^{{10} - {(-2)}} \cdot t^{{4} - {(-6)}} = p^{12}t^{10}$